Ranks of elliptic curves in cyclic sextic extensions of Q

For an elliptic curve $E/Q$ we show that there are infinitely many cyclic sextic extensions $K/Q$ such that the Mordell–Weil group $E\left(K\right)$ has rank greater than the subgroup of $E\left(K\right)$ generated by all the $E\left(F\right)$ for the proper subfields $F\subset K$. For certain curves $E/Q$ we show that the number of such fields $K$ of conductor less than $X$ is $\gg \sqrt{X}$.